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Non-Clifford Gates Are Required for Long-Term Memory

Authors Jon Nelson , Joel Rajakumar , Michael J. Gullans

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LIPIcs.CCC.2026.38.pdf
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Subject Classification

ACM Subject Classification
  • Theory of computation → Quantum information theory
  • Theory of computation → Quantum complexity theory
Keywords
  • Fault Tolerance
  • Quantum Error Correction

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Abstract

We show that all Clifford circuits under interspersed depolarizing noise lose memory of their input exponentially quickly in the depth of the circuit, even when given access to a supply of fresh qubits initialized in arbitrary states. This result applies only in the absence of intermediate measurements, i.e. without intervention by a noiseless external observer such as a classical computer. Nonetheless, this result is surprising given that non-Clifford circuits with access to a supply of fresh qubits can preserve quantum states for arbitrary lengths of time, and even perform fault tolerant quantum computation on them, without requiring any intermediate measurements (Aharonov et al., STOC 1997). Our result shows that such fault tolerance protocols are impossible using only Clifford gates, demonstrating that non-Clifford gates are fundamentally required to store quantum or classical information for long periods of time.

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Jon Nelson, Joel Rajakumar, and Michael J. Gullans. Non-Clifford Gates Are Required for Long-Term Memory. In 41st Computational Complexity Conference (CCC 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 383, pp. 38:1-38:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.CCC.2026.38

Author Details

Jon Nelson
  • Joint Center for Quantum Information & Computer Science, University of Maryland, College Park, MD, USA
  • NIST, Gaithersburg, MD, USA
Joel Rajakumar
  • Joint Center for Quantum Information & Computer Science, University of Maryland, College Park, MD, USA
  • NIST, Gaithersburg, MD, USA
  • IBM T. J. Watson Research Center, Yorktown Heights, NY, USA
Michael J. Gullans
  • Joint Center for Quantum Information & Computer Science, University of Maryland, College Park, MD, USA
  • National Institute of Standards and Technology, Gaithersburg, MD, USA

Funding

This material is based upon work supported by the U.S. Department of Energy, Office of Science, Accelerated Research in Quantum Computing, Fundamental Algorithmic Research toward Quantum Utility (FAR-Qu). Additional support is acknowledged from IBM, where JR perfomed a portion of this research as an intern. This material is based upon work supported in part by the NSF QLCI award OMA2120757. This work was performed in part at the Kavli Institute for Theoretical Physics (KITP), which is supported by grant NSF PHY-2309135. JN is supported by the National Science Foundation Graduate Research Fellowship Program under Grant No. DGE 2236417.

Acknowledgements

We thank Abhinav Deshpande, Kunal sharma, Oles Shtanko, Dominik Hangleiter, Zhi-Yuan Wei, Daniel Malz and Alexey Gorshkov for helpful discussions.

References

  1. Scott Aaronson and Daniel Gottesman. Improved simulation of stabilizer circuits. Phys. Rev. A, 70:052328, November 2004. URL: https://doi.org/10.1103/PhysRevA.70.052328.
  2. D. Aharonov, M. Ben-Or, R. Impagliazzo, and N. Nisan. Limitations of noisy reversible computation, 1996. URL: https://arxiv.org/abs/quant-ph/9611028.
  3. Dorit Aharonov and Michael Ben-Or. Fault-tolerant quantum computation with constant error rate. SIAM Journal on Computing, 38(4):1207-1282, 2008. URL: https://doi.org/10.1137/S0097539799359385.
  4. Dorit Aharonov, Xun Gao, Zeph Landau, Yunchao Liu, and Umesh Vazirani. A polynomial-time classical algorithm for noisy random circuit sampling. In Proceedings of the 55th Annual ACM Symposium on Theory of Computing, STOC 2023, pages 945-957, New York, NY, USA, 2023. Association for Computing Machinery. URL: https://doi.org/10.1145/3564246.3585234.
  5. R. Alicki, M. Horodecki, P. Horodecki, and R. Horodecki. On thermal stability of topological qubit in kitaev’s 4d model. Open Systems & Information Dynamics, 17(01):1-20, 2010. URL: https://doi.org/10.1142/S1230161210000023.
  6. Armando Angrisani, Antonio A. Mele, Manuel S. Rudolph, M. Cerezo, and Zoe Holmes. Simulating quantum circuits with arbitrary local noise using pauli propagation, 2025. URL: https://arxiv.org/abs/2501.13101.
  7. A. Anshu, Y. Liu, Q. Nguyen, and C. Pattison, 2025. Simons talk available at: https://simons.berkeley.edu/talks/quynh-nguyen-harvard-university-2025-05-28. Google Scholar
  8. Dave Bacon, Steven T. Flammia, Aram W. Harrow, and Jonathan Shi. Sparse quantum codes from quantum circuits. In Proceedings of the Forty-Seventh Annual ACM Symposium on Theory of Computing, STOC '15, pages 327-334, New York, NY, USA, 2015. Association for Computing Machinery. URL: https://doi.org/10.1145/2746539.2746608.
  9. Yimu Bao, Soonwon Choi, and Ehud Altman. Symmetry enriched phases of quantum circuits. Annals of Physics, 435:168618, 2021. URL: https://doi.org/10.1016/j.aop.2021.168618.
  10. Michael Ben-Or, Daniel Gottesman, and Avinatan Hassidim. Quantum refrigerator, 2013. URL: https://arxiv.org/abs/1301.1995.
  11. Thiago Bergamaschi and Yunchao Liu. On Fault Tolerant Single-Shot Logical State Preparation and Robust Long-Range Entanglement. In Raghu Meka, editor, 16th Innovations in Theoretical Computer Science Conference (ITCS 2025), volume 325 of Leibniz International Proceedings in Informatics (LIPIcs), pages 16:1-16:9, Dagstuhl, Germany, 2025. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.ITCS.2025.16.
  12. Dolev Bluvstein, Simon J. Evered, Alexandra A. Geim, Sophie H. Li, Hengyun Zhou, Tom Manovitz, Sepehr Ebadi, Madelyn Cain, Marcin Kalinowski, Dominik Hangleiter, J. Pablo Bonilla Ataides, Nishad Maskara, Iris Cong, Xun Gao, Pedro Sales Rodriguez, Thomas Karolyshyn, Giulia Semeghini, Michael J. Gullans, Markus Greiner, Vladan Vuletić, and Mikhail D. Lukin. Logical quantum processor based on reconfigurable atom arrays. Nature, 626(7997):58-65, 2024. Google Scholar
  13. Adam Bouland, Joseph F. Fitzsimons, and Dax Enshan Koh. Complexity Classification of Conjugated Clifford Circuits. In Rocco A. Servedio, editor, 33rd Computational Complexity Conference (CCC 2018), volume 102 of Leibniz International Proceedings in Informatics (LIPIcs), pages 21:1-21:25, Dagstuhl, Germany, 2018. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.CCC.2018.21.
  14. Sergey Bravyi and David Gosset. Improved classical simulation of quantum circuits dominated by clifford gates. Phys. Rev. Lett., 116:250501, June 2016. URL: https://doi.org/10.1103/PhysRevLett.116.250501.
  15. Sergey Bravyi, David Gosset, Robert König, and Marco Tomamichel. Quantum advantage with noisy shallow circuits. Nature Physics, 16(10):1040-1045, July 2020. URL: https://doi.org/10.1038/s41567-020-0948-z.
  16. Michael J. Bremner, Ashley Montanaro, and Dan J. Shepherd. Achieving quantum supremacy with sparse and noisy commuting quantum computations. Quantum, 1:8, April 2017. URL: https://doi.org/10.22331/q-2017-04-25-8.
  17. Nicolas Delfosse and Adam Paetznick. Spacetime codes of clifford circuits, 2023. URL: https://doi.org/10.48550/arXiv.2304.05943.
  18. Eric Dennis, Alexei Kitaev, Andrew Landahl, and John Preskill. Topological quantum memory. Journal of Mathematical Physics, 43(9):4452-4505, September 2002. URL: https://doi.org/10.1063/1.1499754.
  19. Beatriz C. Dias, Domagoj Perkovic, Masudul Haque, Pedro Ribeiro, and Paul A. McClarty. Quantum Noise as a Symmetry-Breaking Field. arXiv preprint, 2022. URL: https://arxiv.org/abs/2208.13861.
  20. Bill Fefferman, Soumik Ghosh, Michael Gullans, Kohdai Kuroiwa, and Kunal Sharma. Effect of nonunital noise on random-circuit sampling. PRX Quantum, 5:030317, July 2024. URL: https://doi.org/10.1103/PRXQuantum.5.030317.
  21. Xun Gao and Luming Duan. Efficient classical simulation of noisy quantum computation, 2018. URL: https://arxiv.org/abs/1810.03176.
  22. Daniel Gottesman. The heisenberg representation of quantum computers, 1998. URL: https://arxiv.org/abs/quant-ph/9807006.
  23. Daniel Gottesman. An introduction to quantum error correction and fault-tolerant quantum computation, 2009. URL: https://arxiv.org/abs/0904.2557.
  24. Daniel Gottesman. Fault-tolerant quantum computation with constant overhead. Quantum Info. Comput., 14(15–16):1338-1372, November 2014. URL: https://doi.org/10.26421/QIC14.15-16-5.
  25. Michael J. Gullans and David A. Huse. Dynamical purification phase transition induced by quantum measurements. Physical Review X, 10(4), October 2020. URL: https://doi.org/10.1103/physrevx.10.041020.
  26. Shao-Kai Jian, Chunxiao Liu, Xiao Chen, Brian Swingle, and Pengfei Zhang. Quantum error as an emergent magnetic field. arXiv preprint, 2021. URL: https://arxiv.org/abs/2106.09635.
  27. Julia Kempe, Oded Regev, Falk Unger, and Ronald de Wolf. Upper bounds on the noise threshold for fault-tolerant quantum computing. In Luca Aceto, Ivan Damgård, Leslie Ann Goldberg, Magnús M. Halldórsson, Anna Ingólfsdóttir, and Igor Walukiewicz, editors, Automata, Languages and Programming, pages 845-856, Berlin, Heidelberg, 2008. Springer Berlin Heidelberg. URL: https://doi.org/10.1007/978-3-540-70575-8_69.
  28. Yaodong Li, Xiao Chen, and Matthew P. A. Fisher. Quantum zeno effect and the many-body entanglement transition. Phys. Rev. B, 98:205136, November 2018. URL: https://doi.org/10.1103/PhysRevB.98.205136.
  29. Yaodong Li, Xiao Chen, and Matthew P. A. Fisher. Measurement-driven entanglement transition in hybrid quantum circuits. Phys. Rev. B, 100:134306, October 2019. URL: https://doi.org/10.1103/PhysRevB.100.134306.
  30. Yue Li and Martin Claassen. Statistical mechanics of monitored dissipative random circuits. Phys. Rev. B, 108:104310, September 2023. URL: https://doi.org/10.1103/PhysRevB.108.104310.
  31. Zhi Li, Shengqi Sang, and Timothy H. Hsieh. Entanglement dynamics of noisy random circuits. Phys. Rev. B, 107:014307, January 2023. URL: https://doi.org/10.1103/PhysRevB.107.014307.
  32. Shuo Liu, Ming-Rui Li, Shi-Xin Zhang, and Shao-Kai Jian. Entanglement structure and information protection in noisy hybrid quantum circuits. Phys. Rev. Lett., 132:240402, June 2024. URL: https://doi.org/10.1103/PhysRevLett.132.240402.
  33. Shuo Liu, Ming-Rui Li, Shi-Xin Zhang, Shao-Kai Jian, and Hong Yao. Universal kardar-parisi-zhang scaling in noisy hybrid quantum circuits. Phys. Rev. B, 107:L201113, May 2023. URL: https://doi.org/10.1103/PhysRevB.107.L201113.
  34. Victor Martinez, Armando Angrisani, Ekaterina Pankovets, Omar Fawzi, and Daniel Stilck França. Efficient simulation of parametrized quantum circuits under nonunital noise through pauli backpropagation. Phys. Rev. Lett., 134:250602, June 2025. URL: https://doi.org/10.1103/j1gg-s6zb.
  35. Antonio Anna Mele, Armando Angrisani, Soumik Ghosh, Sumeet Khatri, Jens Eisert, Daniel Stilck França, and Yihui Quek. Noise-induced shallow circuits and absence of barren plateaus, 2024. URL: https://arxiv.org/abs/2403.13927.
  36. Alexander Müller-Hermes, Daniel Stilck França, and Michael M. Wolf. Relative entropy convergence for depolarizing channels. Journal of Mathematical Physics, 57(2), January 2016. URL: https://doi.org/10.1063/1.4939560.
  37. Jon Nelson, Joel Rajakumar, Dominik Hangleiter, and Michael J. Gullans. Polynomial-Time Classical Simulation of Noisy Quantum Circuits with Naturally Fault-Tolerant Gates, pages 1309-1331. SIAM, 2026. URL: https://doi.org/10.1137/1.9781611978971.50.
  38. Thomas Schuster, Chao Yin, Xun Gao, and Norman Y. Yao. A polynomial-time classical algorithm for noisy quantum circuits, 2024. URL: https://doi.org/10.48550/arXiv.2407.12768.
  39. Oles Shtanko and Kunal Sharma. Complexity of local quantum circuits under nonunital noise, 2024. URL: https://arxiv.org/abs/2411.04819.
  40. Brian Skinner, Jonathan Ruhman, and Adam Nahum. Measurement-induced phase transitions in the dynamics of entanglement. Phys. Rev. X, 9:031009, July 2019. URL: https://doi.org/10.1103/PhysRevX.9.031009.
  41. Fabian Ballar Trigueros and José Antonio Marín Guzmán. Nonstabilizerness and error resilience in noisy quantum circuits, 2025. URL: https://arxiv.org/abs/2506.18976.
  42. Su un Lee, Soumik Ghosh, Changhun Oh, Kyungjoo Noh, Bill Fefferman, and Liang Jiang. Classical simulation of noisy random circuits from exponential decay of correlation, 2025. URL: https://arxiv.org/abs/2510.06328.
  43. John Watrous. cs.uwaterloo.ca. https://cs.uwaterloo.ca/~watrous/QIT-notes/QIT-notes.pdf. [Accessed 11-09-2025].
  44. Mark Wilde. markwilde.com. https://markwilde.com/teaching/2015-fall-qit/lectures/lecture-10.pdf. [Accessed 11-09-2025].
  45. Mithuna Yoganathan, Richard Jozsa, and Sergii Strelchuk. Quantum advantage of unitary clifford circuits with magic state inputs. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 475(2225):20180427, May 2019. URL: https://doi.org/10.1098/rspa.2018.0427.
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